Time-dependent, irreversible entropy production and geodynamics

Abstract

We present an application of entropy production as an abstraction tool for complex processes in geodynamics. Geodynamic theories are generally based on the principle of maximum dissipation being equivalent to the maximum entropy production. This represents a restriction of the second law of thermodynamics to its upper bound. In this paper, starting from the equation of motion, the first law of thermodynamics and decomposition of the entropy into reversible and irreversible terms, 1 we come up with an entropy balance equation in an integral form. We propose that the extrema of this equation give upper and lower bounds that can be used to constrain geodynamics solutions. This procedure represents an extension of the classical limit analysis theory of continuum mechanics, which considers only stress and strain rates. The new approach, however, extends the analysis to temperature-dependent problems where thermal feedbacks can play a significant role. We apply the proposed procedure to a simple convective/conductive heat transfer problem such as in a planetary system. The results show that it is not necessary to have a detailed knowledge of the material parameters inside the planet to derive upper and lower bounds for self-driven heat transfer processes. The analysis can be refined by considering precise dissipation processes such as plasticity and viscous creep.

Thermo‐poro‐mechanics of chemically active creeping faults: 2. Transient considerations

“…In such a regime the system would be allowed to cross steady states, leading to multiplicities of frequencies, transitions from aseismic creep to slow slip or seismic instabilities, and any number of combinations thereof. Once the system is in the oscillatory regime with G r values after the homoclinic point B of Figure a, the global attractor is defined by the critical G r number derived in this contribution, which also corresponds to the point of maximum entropy production of the macroscopic system [ Regenauer‐Lieb et al , ].…”

The effect of pre‐tectonic reaction and annealing extent on behaviour during subsequent deformation: insights from paired shear zones in the lower crust of Fiordland, New Zealand

“…Consequently, predictions of the mechanical behaviour and microstructural response derived from laboratory experiments are potentially inaccurate (e.g. Paterson, , ; Regenauer‐Lieb et al ., ).…”

Continuum damage mechanics for the lithosphere

“…Combining equation (3) with equation (B3) in Appendix B results in the following inequality: Equation (4) contains the fundamental second principle of thermodynamics stating that the rate of entropy production is higher or equal to the heat supply over temperature. Equation (4) represents the local rate of irreversible entropy production as discussed in details by Regenauer‐Lieb et al [2010]. Using the additive decomposition of strain (1) as well as the derivative of Helmholtz free energy with respect to time (∂ x ψ is the partial derivative of ψ with respect to the variable x): , equation (4) reduces to This expression, which invokes the positivity of the irreversible entropy production rate, is often called Clausius‐Duhem inequality.…”

“…This feedback is very efficient for materials with high activation energy such as olivine and it can lead to a substantial reduction in lithospheric strength [ Braeck and Podladchikov , 2007]. However, the predicted level of forces is still an upper limit [ Regenauer‐Lieb et al , 2010]. Additional weakening mechanisms through damage mechanics, temperature or fluid flow must be taken into account.…”